This preview shows page 1. Sign up to view the full content.
Unformatted text preview: ) N (t0 ). The desired
result now follows by induction.
We are now ready for our second deﬁnition. It is in terms of the process
{N (s) : s 0} that counts the number of arrivals in [0, s].
Theorem 2.7. If {N (s), s 0} is a Poisson process, then
(i) N (0) = 0,
(ii) N (t + s) N (s) = Poisson( t), and
(iii) N (t) has independent increments.
Conversely, if (i), (ii), and (iii) hold, then {N (s), s 0} is a Poisson process.
Why is this true? Clearly, (i) holds. Lemmas 2.2 and 2.6 prove (ii) and (iii).
To start to prove the converse, let Tn be the time of the nth arrival. The ﬁrst
arrival occurs after time t if and only if there were no arrivals in [0, t]. So using
the formula for the Poisson distribution
P (⌧1 > t) = P (N (t) = 0) = e t 83 2.2. DEFINING THE POISSON PROCESS
This shows that ⌧1 = T1 is exponential( ). For ⌧2 = T2 T1 we note that P (⌧2 > t⌧1 = s) = P ( no arrival in (s, s + t] ⌧1 = s)
= P (N (t + s) = P (N (t + s) N (s) = 0N (r) = 0 for r < s, N (s) = 1)
N (s) = 0) =...
View Full
Document
 Spring '10
 DURRETT
 The Land

Click to edit the document details